I really enjoyed the pool that came with my apartment in Saigon, and although I am not a big swimmer, I just enjoyed to dip in the pool from time to time. So do many expats, I heard. But not many houses and apartments come with a pool, so you have to rely on the hotels or gyms.
Lao Plaza Hotel, Mercure and Rashmis have all decent pools open to the public for a fee of around 10 USD. Not to bad, but a bit too much for frequent swimmers. If you have a Sengdara fitness center membership, usage of the pool is included. But people told me it is quite crowded on the weekend and not always as clean as expected.
Two more options: The recently opened New Arena (opposite UNICEF turn into the small alley next to the Singaporean Embassy) offers a new (and new is always good in Laos) pool and football fields. Pool fee is 49.000 KIP per day, a coffeeshop/small restaurant is attached. Only disadvantage: Not much shadow at the pool area.
If you don't mind going local, then save money, bring your own food and enjoy the place. It is privately owned, costs 10.000 KIP entrance and is just okay. No fancy features, and I haven't see the toilets from inside, but the water was very clean, and the areas as well. It is located near the Lao Tobacco factory. After the factory (going direction friendship bridge), turn left and then right, follow the street for about 300 meters, the it is on the right side (looks a bit like a regular family home, but the pools are in the back)